Solve 5m+7/4m-6m-2/3m^2+2m+3/6m^3

$$\frac { 5 m + 7 } { 4 m } - \frac { 6 m - 2 } { 3 m ^ { 2 } } + \frac { 2 m + 3 } { 6 m ^ { 3 } }$$

Evaluate

$$\frac{5}{4}-\frac{1}{4m}+\frac{1}{m^{2}}+\frac{1}{2m^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4m and 3m^{2} is 12m^{2}. Multiply \frac{5m+7}{4m} times \frac{3m}{3m}. Multiply \frac{6m-2}{3m^{2}} times \frac{4}{4}.

$$\frac{\left(5m+7\right)\times 3m}{12m^{2}}-\frac{4\left(6m-2\right)}{12m^{2}}+\frac{2m+3}{6m^{3}}$$

Since \frac{\left(5m+7\right)\times 3m}{12m^{2}} and \frac{4\left(6m-2\right)}{12m^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(5m+7\right)\times 3m-4\left(6m-2\right)}{12m^{2}}+\frac{2m+3}{6m^{3}}$$

Do the multiplications in \left(5m+7\right)\times 3m-4\left(6m-2\right).

$$\frac{15m^{2}+21m-24m+8}{12m^{2}}+\frac{2m+3}{6m^{3}}$$

Combine like terms in 15m^{2}+21m-24m+8.

$$\frac{15m^{2}-3m+8}{12m^{2}}+\frac{2m+3}{6m^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12m^{2} and 6m^{3} is 12m^{3}. Multiply \frac{15m^{2}-3m+8}{12m^{2}} times \frac{m}{m}. Multiply \frac{2m+3}{6m^{3}} times \frac{2}{2}.

$$\frac{\left(15m^{2}-3m+8\right)m}{12m^{3}}+\frac{2\left(2m+3\right)}{12m^{3}}$$

Since \frac{\left(15m^{2}-3m+8\right)m}{12m^{3}} and \frac{2\left(2m+3\right)}{12m^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\left(15m^{2}-3m+8\right)m+2\left(2m+3\right)}{12m^{3}}$$

Do the multiplications in \left(15m^{2}-3m+8\right)m+2\left(2m+3\right).

$$\frac{15m^{3}-3m^{2}+8m+4m+6}{12m^{3}}$$

Combine like terms in 15m^{3}-3m^{2}+8m+4m+6.

$$\frac{15m^{3}-3m^{2}+12m+6}{12m^{3}}$$

Factor the expressions that are not already factored in \frac{15m^{3}-3m^{2}+12m+6}{12m^{3}}.

$$\frac{3\left(5m^{3}-m^{2}+4m+2\right)}{12m^{3}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{5m^{3}-m^{2}+4m+2}{4m^{3}}$$

Expand

$$\frac{5}{4}-\frac{1}{4m}+\frac{1}{m^{2}}+\frac{1}{2m^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4m and 3m^{2} is 12m^{2}. Multiply \frac{5m+7}{4m} times \frac{3m}{3m}. Multiply \frac{6m-2}{3m^{2}} times \frac{4}{4}.

$$\frac{\left(5m+7\right)\times 3m}{12m^{2}}-\frac{4\left(6m-2\right)}{12m^{2}}+\frac{2m+3}{6m^{3}}$$

Since \frac{\left(5m+7\right)\times 3m}{12m^{2}} and \frac{4\left(6m-2\right)}{12m^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(5m+7\right)\times 3m-4\left(6m-2\right)}{12m^{2}}+\frac{2m+3}{6m^{3}}$$

Do the multiplications in \left(5m+7\right)\times 3m-4\left(6m-2\right).

$$\frac{15m^{2}+21m-24m+8}{12m^{2}}+\frac{2m+3}{6m^{3}}$$

Combine like terms in 15m^{2}+21m-24m+8.

$$\frac{15m^{2}-3m+8}{12m^{2}}+\frac{2m+3}{6m^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12m^{2} and 6m^{3} is 12m^{3}. Multiply \frac{15m^{2}-3m+8}{12m^{2}} times \frac{m}{m}. Multiply \frac{2m+3}{6m^{3}} times \frac{2}{2}.

$$\frac{\left(15m^{2}-3m+8\right)m}{12m^{3}}+\frac{2\left(2m+3\right)}{12m^{3}}$$

Since \frac{\left(15m^{2}-3m+8\right)m}{12m^{3}} and \frac{2\left(2m+3\right)}{12m^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\left(15m^{2}-3m+8\right)m+2\left(2m+3\right)}{12m^{3}}$$

Do the multiplications in \left(15m^{2}-3m+8\right)m+2\left(2m+3\right).

$$\frac{15m^{3}-3m^{2}+8m+4m+6}{12m^{3}}$$

Combine like terms in 15m^{3}-3m^{2}+8m+4m+6.

$$\frac{15m^{3}-3m^{2}+12m+6}{12m^{3}}$$

Factor the expressions that are not already factored in \frac{15m^{3}-3m^{2}+12m+6}{12m^{3}}.

$$\frac{3\left(5m^{3}-m^{2}+4m+2\right)}{12m^{3}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{5m^{3}-m^{2}+4m+2}{4m^{3}}$$